MA424 Example Sheet 6
2 December 2015
1. Let fr (x) = rx(1 − x) with r ≥ 4. Show that frn has 2n fixed points for
every n ≥ 1.
2. Let G and G0 be the directed graphs associated to the matrices


1 1 0
1 1
AG =
and AG0 = 0 0 1 .
1 0
1 1 0
Show that the associated shift maps on G∞ and G0∞ are topologically
conjugate.
3. Let f : X → X be a continuous map from a compact metric space X
to itself. Define dn (x, y) = max0≤`≤n−1 d(f ` (x), f ` (y)). Show that dn
satisfies the axioms of a metric. Show that dn defines the same topology
on X as d.
4. Show that a compact metric space possesses a finite (n, )-spanning set.
Show that a compact metric space possesses a finite (n, )-separated set.
5. Compute the entropy of the Smale horseshoe.
6. Compute the entropy of a C 2 -diffeomorphism of the circle with an
irrational rotation number.
7. Compute the entropy of the map f : [0, 1] → [0, 1] defined by f (x) =
x(1 − x).
8. Give an example of a continuous map with h(f ) > 0 and no periodic
points.